Finance Theory GroupSummer School

2015

Dynamic Financial Contracting

Prof. Peter DeMarzoStanford University

Part I:Discrete Time Models

Agency Models in Corporate Finance

• Static Agency / Contracting models have proved important in Corporate Finance Capital Structure (Jensen and Meckling, 1976)

• Concentrated equity ownership, Asset substitution

Incentive Schemes (Holmstrom, 1979)• Monotonicity of payoffs, Informativeness of signals

Security Design (Innes, 1990)• Debt contracts, inside equity

2

Why Dynamic Models?

• Key shortcomings of static models Security Design Compensation Capital Structure Shutdown / Termination Investment

• What makes dynamic models challenging? Must balance richness

and tractability (HM 87?)

• Security Design Debt securities are

almost always not single cash flows “min(Y, D).” There are coupon payments, principal amortization, maturity, seniority…

• Compensation A manager’s

compensation is not a single number. It is a path of annual payments and future promises that depend on the firm’s performance

• Capital Structure Firm leverage ratios are

not static, but continually changing over time.

How do we think about credit lines, which are both a source of debt as well as credit (financial slack)?

3

The Plan

• Part I: Discrete-Time Models DeMarzo-Fishman (RFS, 2007a,b)

• “Optimal Long-Term Financial Contracting”• “Agency and Optimal Investment Dynamics”

Methodology, implementation and robust intuitions

• Part II: Continuous-Time Models DeMarzo-Sannikov (JF, 2006)

• “Optimal Security Design and Dynamic Capital Structure in a Continuous-Time Agency Model”

Tractability and deeper insights

4

The Plan

• Part III: Next Steps DeMarzo-Fishman-He-Wang (JF, 2012)

• “Dynamic Agency Theory meets the Q-Theory of Investment”

DeMarzo-Livdan-Tchistyi (2014)• “Risking Other People’s Money: Gambling, Limited Liability,

and Optimal Incentives” DeMarzo-Sannikov (2015)

• “Learning, Termination, and Payout Policy in Dynamic Incentive Contracts”

5

SOME SIMPLE STATIC MODELS

Part I.A:"Begin at the beginning," the King said, very gravely…

6

Static Principal-Agent Problem

• The Problem Risk-neutral and wealthy principal hires an agent to

expend effort or take costly actions Effort / action e affects distribution of outcome s Principal chooses incentive scheme w(s) to motivate

agent

• Frictions Observed outcome s does not perfectly reveal e Agent has limited liability, may be risk-averse

7

Compensation Contracts

• With risk aversion, subject to conditions, optimal wage satisfies (Holmstrom 1979):

MLRP implies increasing wage profile w(s)

• Risk neutrality with limited liability Bang-bang contracts (Innes 1990)

• Minimal payoff below a threshold• Maximum payoff above threshold

Investor monotonicity implies inside levered equity

( | )1

'( ( )) ( | )ef s ea b

u w s f s eMarginal cost of compensation

Likelihood ratio of observed outcome

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Ex Ante vs. Ex Post Actions

• Output depends on actions and noise: s(e,) Standard model: agent chooses action before

observing random shocks

Ex post action model: agent observes random shock prior to choosing action

Note: • in a general dynamic model both may be true!• and in continuous time the distinction may blur altogether…

e s(e,)

e s(e,)

9

Ex Ante vs. Ex Post Actions

• IC constraint Ex ante: maxe E[u(w(s(e,))) – d(e)]

E[u ′ w ′ se ] = de

• Many possible IC contracts; shape set to minimize cost

Ex post: maxe state by state u ′w ′se = de

w ′ = de / (u ′se)

• Shape of contract determined by IC constraint• Level determined by participation constraint / limited liability

Lacker & Weinberg (’89) “Costly State Falsification” Edmans & Gabaix (’08) “Tractable Incentive Contracts”

10

Cash Flow Diversion

• Firm hires manager to generate output Output Y, stochastic Agency Problem

• Manager can divert output• Private benefit of $ per $1 dollar diverted

Contract specifies wage w(Y)

• How can we provideincentives to preventdiversion? w ′(Y) =

yY

y0

yn

y1

⋮

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Cash Flow Diversion

• Optimal contract Incentive Compatibility:

• Manager must receive compensation of per extra dollar of reported output

w(Y) = w + (Y – EY)• Linear contract form (“inside equity”) is independent of the

cash flow distribution or utility function

Limited Liability: w(y0) ≥ 0 E w(Y) ≥ (EY – y0) = Y

• The manager must be exposed to risk – and receive a minimal level of rents – to provide incentives

12

Binary PA Model: Equivalence

• Binary PA setting Risk neutral agent Binary outcomes (H vs. L) Ex ante effort (work/shirk)

• Working raises probability of high output from p to p + • Working has private cost of c (H – L)

IC constraint: (wH – wL) ≥ c(H – L) w/Y ≥ c/

• Binary PA Cash Flow Diversion with = c/ Continuous time PA model “looks like” cash flow

diversion model

Y

L

Hp

1-p

13

DISCRETE TIME DYNAMIC AGENCY

Part I.B:"…and go on till you come to the end…"

14

Multiple Periods

• Discrete time model Independent output Yt each period Contract sets payment w based on output history Contract may also terminate; liquidation value Lt

• Repeat static contract (IC): wt = (Yt – yt0) Expected rent = PV(t …)

Y1 Y2 YT…

L1 L2 LT

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A

P

A

P

LT-1

• What payoff pairs are feasible for a risk-neutral principal and agent?

Contract Curve

Y1 Y2 YT…

L1 L2 LT

A

P

LT =0

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Dynamic Contract

• Intuition Limited Liability + IC:

Agent must earn rents Impatience:

Pay cash beyond some threshold

Deferred compensation: Use past payments to “buy” future continuation rents

A

P

L

17

Many Periods (DF 2007)

= 0.5Y = {0,2} = = 1r = 10% = 0.5%

First Best

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5Agent's Payoff a

0

1

2

3

4

5

6

7

8

9

Inve

stor

s' P

ayof

f b

PROPOSITION 4. (OPTIMAL CONTINUATION FUNCTION) Given a0

t and bt concave, the continuation function at s t is given by a0

s Rs and

0

ˆ ( ) if( )( ) if

Ls s

s Ls s s s s

b a a ab aL l a R a a a

, (13)

where

1 0inf : ( ) 1t t ta a a b a , (14)

0 1

1

1 1 1

( ) for ( )

( ) ( ) for t t t

t

t t t t

b a a a ab a

b a a a a a

, (15)

0 ( ) 0ˆ t ss t ta e a , (16)

( ) 1 ( )ˆ ( ) ( )r t s t ss t t t tb a e E b e a Y , (17)

0ˆ ( ) ˆsup : max ,s s

s s s

s

b a Ll a a Ra R

, and (18)

0 ˆˆinf max , : '( )L Ls s s s s sa a a R b a l . (19)

Note finally that bs is concave. 18

5 10 15 20 25 30 35 40 45Agent's Payoff a

00

10

20

30

40

50

60

70

80

90

Inve

stor

s' P

ayof

f b

‐10 ‐5 0 5 10 15 20 25 30

Volatility and Skewness

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Many Periods (DF 2007)

= 0.5Y = {0,2} = = 1r = 10% = 0.5%

First Best

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5Agent's Payoff a

0

1

2

3

4

5

6

7

8

9

Inve

stor

s' P

ayof

f b

PROPOSITION 4. (OPTIMAL CONTINUATION FUNCTION) Given a0

t and bt concave, the continuation function at s t is given by a0

s Rs and

0

ˆ ( ) if( )( ) if

Ls s

s Ls s s s s

b a a ab aL l a R a a a

, (13)

where

1 0inf : ( ) 1t t ta a a b a , (14)

0 1

1

1 1 1

( ) for ( )

( ) ( ) for t t t

t

t t t t

b a a a ab a

b a a a a a

, (15)

0 ( ) 0ˆ t ss t ta e a , (16)

( ) 1 ( )ˆ ( ) ( )r t s t ss t t t tb a e E b e a Y , (17)

0ˆ ( ) ˆsup : max ,s s

s s s

s

b a Ll a a Ra R

, and (18)

0 ˆˆinf max , : '( )L Ls s s s s sa a a R b a l . (19)

Note finally that bs is concave. 20

Implementation (DF 2007)

Implementing the Optimal Contract Long Term Debt: A long term debt contract is characterized by a

sequence of fixed payments xt . If a payment is not made, the firm is in default.

Credit Line: A credit line is characterized by an interest rate r̂ and a fixed credit limit cL

t 0. No payments need be made on the credit line except for required interest payments once the limit is reached, required payments if the limit is reduced.

If not paid, the firm is in default.

Default: Given default on payments totaling z 0 in period t, the investor terminates with probability

ptzz Nt keeping the proceeds Lt.

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Sample Dynamics

0 5 10 15 200

5

10

15

20

25

30

35

Horizon

• Debt and Credit Limit vs. Horizon ( = 10.1%, 10.5%, 15%)

debt coupons

credit limit

22

RECURSIVE CONTRACTINGAND INVESTMENT DYNAMICS

Part I.C:

“If you don’t know where you are going, any road will take you there…”

23

Recursive Contracting

• In a dynamic model, the agent can be compensated in two ways Direct cash payments Promises of future payments (“continuation value”)

• Paying via continuation value relaxes future IC constraints If the agent is not impatient, compensation should be

maximally deferred• Dynamic Programming approach to solve for optimal

contract Abreu-Pearce-Stacchetti (1986), Spear-Srivastava (1987), Phelan-

Townsend (1991), Atkeson (1991), Ljungqvist-Sargent (2000)24

Recursive Contracting

• Methodology: Describe contract recursively using promised future payoffs as state variables Investor’s value function:

• b(w) = max payoff to investor from incentive compatible contract that provides payoff w to the agent

• Optimal Contract Provides expected payoffs (wt, b(wt)) to agent and investors Based on today’s reported cash flows Yt, contract specifies

• Transfers between agent and investors• Tomorrow’s contract: (wt+dt, b(wt+dt))• Probability of Termination

• Given the value function b, this is a static problem

provide incentives

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A General Framework

• Investor value bt(w) Concave (randomize) b’(w) ≥ -1 (pay cash)

• General agency problem Max E[Yt + bt(wt) | et] s.t.

(IC) et argmax E[wt – ct | et ](PC) E[wt – ct | et ] = wt-

(LL) wt ≥ 0

• Monotonicity of wt(Yt,wt-) Cash payouts follow high

cash flows

FeasibleIC Contracts

Agent w

Investors b

first best

b(w)

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Investment Dynamics

• Scalable Technology Investment option

• Rescales payoffs by • Capital & adj. cost c()• (w, b) (w, b – c())

Optimal growth• b-(w) = max b+(w/ ) – c()• b concave with w

• Monotonicity of wt(Yt,wt-) Investment increases with

current & past cash flow Investment increases with

past investment

c()

downsize< > expand

Agent w

Investors b

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Conclusions

• Dynamic Financial Contracting Compensate the agent with cash, or with future

promises (“continuation value”) Deferred compensation provides future financial slack Relatively simple capital structure may capture

complicated contract dynamics Agency concerns generally lead to capital structure

path dependence & positive feedback in investment dynamics

• What’s Next? Not yet sufficiently tractable Continuous-time will simplify & allow for new insights

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